An Efficient Bayesian Framework for Inverse Problems via Optimization and Inversion: Surrogate Modeling, Parameter Inference, and Uncertainty Quantification

2026-02-04

AI Summary

This work proposes a novel Bayesian framework that integrates Bayesian optimization with Bayesian inversion to address inverse problems in scenarios where high-fidelity models are computationally expensive and observational data are limited. By adaptively constructing a Gaussian process surrogate model, the method jointly leverages prior information and observed data to achieve efficient parameter inference and robust uncertainty quantification. The key innovation lies in the synergistic coupling of Bayesian optimization and Bayesian inversion, which significantly enhances both the efficiency of surrogate model construction and the accuracy of the inversion. Experimental results on multiple benchmark test functions demonstrate that the proposed framework achieves high-precision parameter estimates and reliable uncertainty quantification while substantially reducing computational cost, thereby offering an efficient and trustworthy tool for engineering decision-making.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Sampling/Simulation-based Search
• Machine Learning: Bayesian Learning

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

The present paper proposes a Bayesian framework for inverse problems that seamlessly integrates optimization and inversion to enable rapid surrogate modeling, accurate parameter inference, and rigorous uncertainty quantification. Bayesian optimization is employed to adaptively construct accurate Gaussian process surrogate models using a minimal number of high-fidelity model evaluations, strategically focusing sampling in regions of high predictive uncertainty. The trained surrogate model is then leveraged within a Bayesian inversion scheme to infer optimal parameter values by combining prior knowledge with observed quantities of interest, resulting in posterior distributions that rigorously characterize epistemic uncertainty. The framework is theoretically grounded, computationally efficient, and particularly suited for engineering applications in which high-fidelity models -- whether arising from numerical simulations or physical experiments -- are computationally expensive, analytically intractable, or difficult to replicate, and data availability is limited. Furthermore, the combined use of Bayesian optimization and inversion outperforms their separate application, highlighting the synergistic benefits of unifying the two approaches. The performance of the proposed Bayesian framework is demonstrated on a suite of one- and two-dimensional analytical benchmarks, including the Mixed Gaussian-Periodic, L\'evy, Griewank, Forrester, and Rosenbrock functions, which provide a controlled setting to assess surrogate modeling accuracy, parameter inference robustness, and uncertainty quantification. The results demonstrate the framework's effectiveness in efficiently solving inverse problems while providing informative uncertainty quantification and supporting reliable engineering decision-making at reduced computational cost.

Problem

• inverse problems
• surrogate modeling
• parameter inference
• uncertainty quantification
• Bayesian framework

Innovation

• Bayesian optimization
• Bayesian inversion
• Gaussian process surrogate
• uncertainty quantification
• inverse problems